Twin-width of graphs with tree-structured decompositions

TL;DR

This paper explores the relationship between twin-width and tree-structured decompositions of graphs, establishing bounds based on various decomposition parameters and analyzing how these influence twin-width.

Contribution

It provides new bounds on twin-width in terms of tree-width and decomposition components, advancing understanding of twin-width's behavior in structured graphs.

Findings

Twin-width is at most twice the strong tree-width.

Linear bound on twin-width given biconnected component widths.

Quadratic bound for quasi-4-connected components.

Abstract

The twin-width of a graph measures its distance to co-graphs and generalizes classical width concepts such as tree-width or rank-width. Since its introduction in 2020 (Bonnet et. al. 2020), a mass of new results has appeared relating twin width to group theory, model theory, combinatorial optimization, and structural graph theory. We take a detailed look at the interplay between the twin-width of a graph and the twin-width of its components under tree-structured decompositions: We prove that the twin-width of a graph is at most twice its strong tree-width, contrasting nicely with the result of (Bonnet and D\'epr\'es 2022), which states that twin-width can be exponential in tree-width. Further, we employ the fundamental concept from structural graph theory of decomposing a graph into highly connected components, in order to obtain an optimal linear bound on the twin-width of a graph given the widths of its biconnected components. For triconnected components we obtain a linear upper bound if we add red edges to the components indicating the splits which led to the components. Extending this approach to quasi-4-connectivity, we obtain a quadratic upper bound. Finally, we investigate how the adhesion of a tree decomposition influences the twin-width of the decomposed graph.